No escaping Einstein. Car batteries would have far lower voltage if not for the relativistic affects in the lead atom, according to simulations.×

You don’t need a near-light-speed spaceship to see the effects of relativity–they can arise even in a slow-moving automobile. The lead-acid battery that starts most car engines gets about 80 percent of its voltage from relativity, according to theoretical work in the 7 January Physical Review Letters. The relativistic effect comes from fast-moving electrons in the lead atom. The computer simulations also explain why tin-acid batteries don’t work, despite apparent similarities between tin and lead.

Electrons typically orbit their atoms at speeds much less than the speed of light, so relativistic effects can largely be ignored when describing atomic properties. But notable exceptions include the heaviest elements in the periodic table. Their electrons must orbit at near light speed to counter the strong attraction of their large nuclei. According to relativity, these high-energy electrons act in some ways as though they have greater mass, so their orbitals must shrink in size compared with slower electrons to maintain the same angular momentum. This contraction, which is most pronounced in the spherically-symmetric s-orbitals of heavy elements, explains why gold has a yellowish hue and why mercury is liquid at room temperature [1].

Previous work has studied the relativistic effects on lead’s crystal structure, but little research has been done on this heavy element’s chemical properties. So Rajeev Ahuja of Uppsala University in Sweden and his colleagues decided to study the most ubiquitous form of lead chemistry: the lead-acid battery. This 150-year-old technology is based on cells consisting of two plates–made of lead and lead dioxide (PbO2)–immersed in sulfuric acid (H2SO4). The lead releases electrons to become lead sulfate (PbSO4), while the lead dioxide gains electrons and also becomes lead sulfate. The combination of these two reactions results in a voltage difference of 2.1 volts between the two plates.

Although theoretical models of the lead-acid battery already exist, Ahuja and his collaborators are the first to derive one from fundamental physics principles. To find the cell’s voltage, the team calculated the energy difference between the electron configurations of the reactants and the products. As with textbook physics problems involving balls rolling down hills, there was no need to simulate the details of intermediate states, as long as the initial and final energies could be calculated.

“The really difficult part is simulating the sulfuric acid electrolyte,” says team member Pekka Pyykkö of the University of Helsinki. To avoid it, the researchers imagined that the reaction started not with the acid, but with the creation of the acid from SO3, which is easier to simulate. At the end they subtracted the energy for the acid creation (known from previous measurements) from the total. By switching relativistic parts of the models “on” and “off” the team found that relativity accounts for 1.7 volts of a single cell, which means that about 10 of the 12 volts in a car battery come from relativistic effects.

Without relativity, the authors argue, lead would act more like tin, which is just above it in the periodic table and which has the same number of electrons (four) in its outermost s- and p-orbitals. But tin’s nucleus has only 50 protons, compared with lead’s 82, so the relativistic contraction of tin’s outermost s-orbital is much less. Additional simulations showed that a hypothetical tin-acid battery would produce insufficient voltage to be practical, because tin dioxide does not attract electrons strongly enough. Tin’s comparatively loose s-orbital does not provide as deep an energy well for electrons as lead does, the team found. In the past, researchers only had a qualitative understanding of why tin-acid batteries never worked out.

Ram Seshadri of the University of California, Santa Barbara, says that relativistic effects were expected, but he had no idea that they would be so dominant. “On the scope of the work, the ability to reliably simulate so complex a device as a lead-acid battery from (almost) first-principles, including all relativistic effects, is a triumph of modeling,” Seshadri says.

–Michael Schirber

Michael Schirber is a Corresponding Editor for Physics based in Lyon, France.